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Tag Archives: modules
Commutative Algebra 58
We have already seen two forms of unique factorization. In a UFD, every nonzero element is a unique product of irreducible (also prime) elements. In a Dedekind domain, every nonzero ideal is a unique product of maximal ideals. Here, we … Continue reading
Posted in Advanced Algebra
Tagged annihilators, associated primes, localization, module division, modules, supports
4 Comments
Commutative Algebra 35
Noetherian Modules Through this article, A is a fixed ring. For the first two sections, all modules are over A. Recall that a submodule of a finitely generated module is not finitely generated in general. This will not happen if we constrain … Continue reading
Posted in Advanced Algebra
Tagged flat modules, hilbert basis theorem, ideals, modules, noetherian, projective modules
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Commutative Algebra 28
Tensor Products In this article (and the next few), we will discuss tensor products of modules over a ring. Here is a motivating example of tensor products. Example If and are real vector spaces, then is the vector space with … Continue reading
Posted in Advanced Algebra
Tagged bilinear maps, distributive property, modules, tensor product, universal properties
2 Comments
Commutative Algebra 21
Exact Sequences When studying homomorphisms of modules over a fixed ring, we often encounter sequences like this: where each is a homomorphism of modules. This sequence may terminate (on either end) or it may continue indefinitely. Note on indices: usually … Continue reading
Posted in Advanced Algebra
Tagged additive functors, exact sequences, functors, modules, short exact sequences
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Commutative Algebra 10
Algebras Over a Ring Let A be any ring; we would like to look at Amodules with a compatible ring structure. Definition. An –algebra is an module , together with a multiplication operator such that becomes a commutative ring (with 1); multiplication … Continue reading
Posted in Advanced Algebra
Tagged algebras, generated submodules, homomorphism, modules, quotient modules, rings, submodules
5 Comments
Commutative Algebra 9
Direct Sums and Direct Products Recall that for a ring A, a sequence of Amodules gives the Amodule where the operations are defined componentwise. In this article, we will generalize the construction to an infinite collection of modules. Throughout this article, let denote … Continue reading
Posted in Advanced Algebra
Tagged direct products, direct sums, modules, rings, universal properties
5 Comments
Commutative Algebra 8
Generated Submodule Since the intersection of an arbitrary family of submodules of M is a submodule, we have the concept of a submodule generated by a subset. Definition. Given any subset , let denote the set of all submodules of M containing … Continue reading
Posted in Advanced Algebra
Tagged free modules, generated submodules, homomorphism, modules, quotient modules, rings, submodules
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Commutative Algebra 7
Modules Having dipped our toes into algebraic geometry, we are back in commutative algebra. Next we would like to introduce “linear algebra” over a ring A. Most of the proofs should pose no difficulty to the reader so we will … Continue reading
Posted in Advanced Algebra
Tagged ideals, linear algebra, module homomorphism, modules, rings, submodules
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Idempotents and Decomposition
Let R be a general ring, not necessarily commutative. An element x∈R is said to be idempotent if x2 = x. Note An endomorphism f of an Rmodule M (i.e. ) is an idempotent if and only if f is a projection, i.e. M = ker(f) ⊕ im(f) and f … Continue reading
Posted in Notes
Tagged blocks, idempotents, indecomposable modules, modules, primitive idempotents
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Tensor Product over Noncommutative Rings
Following the earlier article on tensor products of vector spaces, we will now look at tensor products of modules over a ring R, not necessarily commutative. It turns out we have to distinguish between left and right modules now. Indeed recall … Continue reading
Posted in Notes
Tagged bimodules, hom functor, leftexact, modules, rightexact, tensor products
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